In the cost function below, \( C(x) \) is the cost of producing \( x \) items. Find the average cost per item when the required number of items is produced. \( C(x)=4.5 x+9,300 \) \( \begin{array}{lll}\text { a. } 200 \text { items } & \text { b. } 2000 \text { items } & \text { c. } 5000 \text { items }\end{array} \)

To find the average cost per item when the required number of items is produced, we need to calculate the average cost by dividing the total cost by the number of items produced. Given: \[ C(x) = 4.5x + 9,300 \] We need to find the average cost per item for the following scenarios: a. 200 items b. 2000 items c. 5000 items Let's calculate the average cost for each scenario: a. For 200 items: \[ \text{Average Cost} = \frac{C(200)}{200} \] b. For 2000 items: \[ \text{Average Cost} = \frac{C(2000)}{2000} \] c. For 5000 items: \[ \text{Average Cost} = \frac{C(5000)}{5000} \] Now, let's calculate the average cost for each scenario. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(4.5\times 200+9300\right)}{200}\) - step1: Remove the parentheses: \(\frac{4.5\times 200+9300}{200}\) - step2: Multiply the numbers: \(\frac{900+9300}{200}\) - step3: Add the numbers: \(\frac{10200}{200}\) - step4: Divide the terms: \(51\) a. For 200 items, the average cost per item is $51. Now, let's calculate the average cost for 2000 items. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(4.5\times 2000+9300\right)}{2000}\) - step1: Remove the parentheses: \(\frac{4.5\times 2000+9300}{2000}\) - step2: Multiply the numbers: \(\frac{9000+9300}{2000}\) - step3: Add the numbers: \(\frac{18300}{2000}\) - step4: Reduce the fraction: \(\frac{183}{20}\) b. For 2000 items, the average cost per item is $9.15. Finally, let's calculate the average cost for 5000 items. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(4.5\times 5000+9300\right)}{5000}\) - step1: Remove the parentheses: \(\frac{4.5\times 5000+9300}{5000}\) - step2: Multiply the numbers: \(\frac{22500+9300}{5000}\) - step3: Add the numbers: \(\frac{31800}{5000}\) - step4: Reduce the fraction: \(\frac{159}{25}\) c. For 5000 items, the average cost per item is $6.36. Therefore, the average cost per item for the given scenarios is: a. $51 for 200 items b. $9.15 for 2000 items c. $6.36 for 5000 items